A curve interpolation of a numerical control device or the like, especially an involute interpolation, is very important when machining gears, and pump blades, etc., and therefore, in general, an involute curve is interpolated by the numerical control device and another computer or an NC program generator, and decomposed into linear data, and a numerical control machining is executed by using a tape containing the data.
In view of the above, the present applicants filed an involute interpolation speed control system, as Japanese Patent Application No. 62-157302 (Japanese Laid-Open Patent Publication No. 64-2106), in which an involute curve is easily interpolated in a numerical control device so that the tangential velocity thereof is constant regardless of the angle.
In this involute interpolation speed control system, the coordinates of a point on the involute curve are given by EQU X=R{ cos (.THETA.+.THETA.1)+.THETA. sin (.THETA.+.THETA.1)+X.sub.o, EQU Y=R{ sin (.THETA.+.THETA.1)+.THETA. cos (.THETA.+.THETA.1)+Y.sub.o. EQU .THETA. ranging from .THETA.=(.THETA.2-.THETA.1) to .THETA.=(.THETA.3-.THETA.1)
is incremented by EQU .THETA..sub.n+1 =.THETA..sub.n +K/(R.multidot..THETA.),
and a point (X.sub.n+1, Y.sub.n+1) corresponding to this increment is obtained from the above equations, and the difference between this point and the preceding point is obtained, whereby the involute curve is interpolated.
Here R is the radius of a basic circle, and X.sub.o and Y.sub.o are the coordinates of the center of the basic circle.
Thus, the interpolation is made so that the tangential velocity is constant by setting the increment of .THETA. at a value, K/(R.multidot..THETA.), such that the increment is reduced in inverse proportion to the increase of the angle.
According to the conventional involute interpolation speed control system, however, a bite or leftover occurs in a region where the curvature radius of the involute curve is relatively short, such as the region in the vicinity of the basic circle, due to a servo response delay or thermal deformation of the workpiece.
FIG. 2 is a diagram showing the manner of a machining based on the conventional involute interpolation. In FIG. 2, the basic circle C is a circle which forms the basis of the involute curve. The center O of the basic circle C is given by coordinates (X.sub.o, Y.sub.o), and its radius is given by R.
A point Ps1 is the start point of an involute curve In1, and a point Pe1 is the end point thereof; a point As1 is the start point of an arcuate curve A1, and a point Ae1 is the end point thereof; a point As2 is the start point of an arcuate curve A2, and a point Ae2 is the end point thereof; a point Ps2 is the start point of an involute curve In2, and a point Pe2 is the end point thereof. The respective positional coordinates of these points and the like are previously given as commands to the numerical control device by using a tape or the like.
A tool W is moved interpolatively following a series of program command paths consisting of the involute curve In1, arcuate curve A1, arcuate curve A2, and involute curve In2, but if a machining is actually effected in accordance with this program, the tool W interpolates a path such as the one indicated by dotted line Re, and thus the workpiece is machined to a shape obtained by shaving off the hatched portion from a command work shape, i.e., a shape subject to the bite.
This bite starts at a point Ps3 at a distance corresponding to a curvature radius Rs from the basic circle C, and its depth gradually increases towards the point Pe1. At the point Pe1, the bite depth is equal to a distance De in the normal direction of the involute curve In1. When interpolating the arcuate curve A1, after the end of the interpolation of the involute curve In1, the bite depth gradually decreases toward the point Ae1.
Since this bite is produced at those portions which require a high-accuracy machining, i.e., a small-radius portion of the involute curve In1 and that portion of the arcuate curve A1 which constitutes a joint, it is a serious problem when carrying out the interpolation machining of involute curves by a numerical control device.
The bite effect described above is produced when a convex portion of the involute curve is machined. In contrast, a leftover occurs at a concave portion of the involute curve, and thus the same problem arises.